Automatic solution of linear simultaneous equations



AUTOMATIC SOLUTION OF LINEAR SIMULTMEOUS EQUATIONS Filed Nov. 15, 1946 2 Sheets-Sheet 1 Snoentor fieogge 60; firown I (Knox-neg G. W. BROWN Dec. 14, 1948.

AUTOMATIC SOLUTION 0F LINEAR SIMULTANEOUS EQUATIONS 2 Sheets-Sheet 2 Filed Nov. 15, 1946 Zhmcntor 56o eldfimwz Patented Dec. 14, 1948 AUTOMATIC SOLUTION OF LINEAR SIMULTANEOUS EQUATIONS George W; Brown, Ames, Iowa, assignor to Radio Corporation of America, a corporation of Delaware - Application November 15, 1946, Serial No. 709,920

This invention relates to/'8 method of and means for electronically-solving. linear simultaneous equations.

In a copending applicationofG. W. Brown and E. A; Goldberg, SerialNo, 690,865,.filed August 16,

1946, for fElectronic equation solverj, there is described a method of and means for electronically solvingv non-singular linear. simultaneous equations whose characteristic roots, while complex, have positive real parts. The primary object of this invention is to overcome this limitation of the earlier system and to ensure the stable solution of non-singular linear equations by electronic feedback methods independently of the nature of the equation andwithout dependence for stability on the particular equation to be solved. Otherwise stated, the present invention provides universal stability in a .feedbacktypeof electronic simultaneous equation solver for all systems .of non- Where X1, X2 Xn are the unknown terms implicitly determined by the .n'equations;

' Y1, Y2 Yn are the constant terms of the n equations; 1

an, an anrarethe coeificients, of X1 in the first, second to nth equations, respectively;

an, (L22 (Z112 are the .coefficients of X2 in the first, second, to the, nth equations, respec-= tively; and

am, can cm are the coeffi-cients of Xn in the first, second, to nthequations, respectively.

It should benoted that in the terminology herein employed the first'subscript of each so-l efiicient a identifies the equation in which it is found while the second subscript identifies the particular unknown term X which it modifies.

The simultaneous equations above may be expressed in an equivalent well known matrix form-.

1 n. n ll] I?) (1 la I. ll z Xig Y .Y e X.

' 14 Claims. (Cl. 235-61) 2 which says that when the vector whose componentsare X1, X2 X11 is operated on by the matrix of coeflicients a the result is a new vector Whose n components are expressed in Equations (1),, (2) (n) above. A further simplification is to let the notation Y=AX (5) express the most general form of the matrix equatiorr In this form A is understood to represent a double series of co-efiicients a to be applied to the series of unknown terms X in a group of equations in which Y represents the set of constant terms.

When a vector is operated on by a transposed matrix the coefficients a are interchanged in rows and columns. This operation is indicated by the term A. When a vector is operated on by a matrix A, and the result is operated on by another matrix B, there is an equivalent matrix, denoted by BA, which represents the composite operation.

There is a mathematical theorem, which need not be proved herein, that a matrix of the form A A, corresponding to the step of operating on a vector by A and then operating on the resultant by A, has the property that all its roots are always real and positive. That is, A A is always positive definite. Ihus, if equations AX=Y are replaced by the set AtrAX AtrY (6) having the same solutions as the original set, the characteristic roots of the new system are all positive.

Since the equation solver disclosed in the originalsdisclosure was limited to the solution of linear simultaneous equations whose characteristic roots, while complex, had positive real parts, it follows that a system which expresses the original equations in an equivalent form involving the transposed matrix operation, having positive real roots, will consequently be universally stable, provided there is a unique solution at all and provided further that there is no inherent instabiity in the circuit aside from the nature of the matrix equation. a

It'is, therefore, a further object of this invention to provide a method of and means for electronically solving a non-singular system of simultaneous equations. It is a further object of this invention to provide a method of and means for electronically solving a non-singular system of simultaneous equations independently of the nature oi the characteristic roots of the system,

A further object of this invention is to provide an improved electronic simultaneous equation solver.

A still further object of this invention is to provide an electronic simultaneous equation solver having universal stability.

A stillrfurthersobject is to provide an improved method of "determining electronically the unknown terms of a system of non-singular simultaneous equations.

A still further object is to overcome theed-isadvantages and limitations of previously known. systems for electronically solving a non-singular system of simultaneous equations.

The novel features that are considered char-- acteristic of this invention are set forth with particularity in the appended. claims. Theainvention itself, however, both as to its organization and method of operation, as well as additional objects and advantages thereof, will best be-understood from the following description when read in connection with theaccompanying drawings,- in which I Figure 1 is a-block diagram of an electronic equation solver in accordance with this invention;

Figure 2 is the circuit diagram of the networks used in Fig. 1, and

Figure '3 is a-block diagram of an equation solver of the earliertype to illustrate how it may be-used-"in practicing this invention.

Referring toFig. 1, asignal generator I provides-at terminals 3 and 5 a pair of voltages which are equal in amplitude and of opposite phase measured with respect toground. This may be a D. C. generator, but because of convenience in ampl-ifying it is preferable to use an alternating voltagehaving any suitable frequency, say 1000 cycles, for example. A connection bus 1, 1a distributes the voltage toeach polarity reversing and potentiometerdevice it'being understood that there will be as many of these devices as there -are-equations to be solved.

For convenience the polarity reversing and potentiometer device is hereincalled a network. Its function is to make available at its output terminal by means of a suitable switch avoltage ofone phase or the other as desired, having an amplitude which isa desired known fraction of the amplitude of the voltage applied to its input.

-S ince all networks 9 are identical the detailed construction has been shown but once in Fig. 2, to which'reference is now made.

Each network includes a double-pole doublethrow switch ll, an accurately calibrated patch-" 2.net

ti'ometer 13 having an overall resistance of, say, 1000 ohms, and a'fixed resistor [5 of equal resistance. The potentiometer l3 and resistor 15 are connected between the two movable switch armsand ground while the switch contacts are connected oppositely to the bus conductors '1, la .sothat in one position of the switch the potentiometer is'connected between conductor 1 and ground while the resistor is connected between conductor 1a and ground, and the connectionsare'reversed in the other position of the switch.

Since the accuracy of the system depends to a large degree upon theaccuracy with whichthe "potentiometers can be set, they are preferably of the type-consisting of a -turn helical resistance and are equipped with a counter and a Vernier 'dial'whicli enables the voltage division to be set accurately to 3 significant figures. The fixed resistor is used to load the conductor which is not in use'equally with the one-which is in use to pre- 4 vent the operation of the switch from changing the voltage by reason of a change in load impedance.

Returning again to Fig. 1, it should be noted 5 that thenetworks so far described are used to establish voltages representing the constant terms Y1','j.Y2". ...-Ya 6f the-equations; To vcorrelate "the apparatus with the+equations each component is identified by the symbol of the term which lO -i't represents, as is the voltage produced thereby.

.-Thus the .Y 1. network is set by means of the phastingtsw'itch-andpotentiometer so that the output voltage Y1 produced thereby is proportionately 'related'to-"the' input voltage as determined by the 'constant' term yr'of the equation. Of course, if the input voltage is equal to a power of 10 then the :output yoltage-Xr may readily be made equal to the. constant term. However, this is not necessary' as will be discussed later.

The output voltages. of the networks are ap- 'plied throughresistors 111,- I12 :lliito therespective input terminalsvof alike number ofiidentical input amplifiers I91, I92 l9n,.identified asamplifiers' Xi',"-' -X2:. ."Xn' and each having a gaincpf. 'I' he input. amplifiers have push-pull output terminals to which -zare connected conductors =2l1, 212:. 2hie-each 'pair. beingconnected tothe input' of-*nnetworks 9. For convenience again; I these networks-are identified: by the coefficient term w-hichethey. represent but these networks are all primed since *the rows and columns area-transposediwi'th respect: to the conventio'n' oflnotationaadoptedabove. Thus amplifier 'Xi supplies the voltage :Xflto networks 11112 1112,- Ja-m,1thefirst:subscriptindicating that 'these networks @modify cth'e output avoltage Xi. SimilarlyJWith -the other networks.

Additionalomput amplifiers 231, 232 23!). are -=provided,;"theoutput voltages of these ampli- 4o 'fiers representing the -unknown terms of the n equations. =The amplifiers are a similarly identified -as"-*X1', Xz, ."-Xm and havlng a-gain of a. Each outputamplifier' is coupled to one network ofeach iinput amplifier; Specifically, the output terminals of'the' primed or transposed-networks are" connected? through"resistors-in each-case, t0 the input of theamplifier identified-bythe sec- .ond number of thenetworksubscriptt Thus networks a11 ',a21' am connect, through re- -si'stors 25, -21:'ar1d 29*respectivelyg'torthe input of amplifier X1;,, networks; a12',1,a22,. anz connect, through resistors 3|, 33 and 35, respectively, to= theinput' of amplifier and networks am, am .lanil" connect-,".through"= resistors: 31, 39

4 I respectively; to th'einputofamplifier Xn.

Each output amplifier X1, X2 Xais-connected through' ione of th'e bus connectors 431, A32, Q 435 respectively; to: the input of n net- "works -9; identified bythe coefficient terms which "they-"represent,1-or, otherwise stated, which produce voltages representative of the coefficients of X of the n equations: Thus output amplifier X1 feeds-into=netwcrks an, a-2 1 amyamplifier'Xz feedsintonetworks amam, .dnz; and ampliher Xn feeds' -intometworksan, azn 'Ctnu.

*The last group of networks-feeds into the'first -or primed set- -of-wamplifiers; In this case, however, the connections are reversed 'from those I used-to"connect'thefirstnetworks. Thus the net works" of the 'secorid'group are 'connected'through resistors. in each case, to the input of the input or primed amplifier identified lby ,cthe first numher of thernetwlork subscript. ;'?I:hus, network an, an andsam connect, through resistors 45, 41 and 49 respectively; to the input of amplifier X1; net- Similarly,

works an, (122, and (121i connect, through resistor 53;- and 'SFrespectiVeIy, to the input of amplifier .Xn be the output voltages fiers bearing the sameidentification and values are available); letEr, E2 En be the residual voltages at the input terminals of amplisubstituting (11) and (12) in (7), (8) and (9) factoring n; and rearranging j gi l2 Ext- 22 32+ 112 n] (17) %"=ia..fEi+az.'E amen] 18 By design the product ,u'p. is made very large,

' say of the order of ten to fifteen thousand, so that the terms can be considered as being substantially zero and the right hand portion of each equation (16), (17) and (18) is then equal to zero.

- In the matrix notation, the n equations (7) to Finally, substituting 21) m (227 l X= A"[AX+ Yl 1 (23) ,Now, if A is itself non-singular it must be true that Thus, if

Hence X is the required solution.

For the stability analysis, consider Equation 23. This is analogous to the equation developed in the Brown-Goldberg patent application referred to above, in which A A replaces A, A' Y replaces Y and I-W M I +1) replaces n+1 Thus, for purposes of stability the characteristic roots of A A are relevant now, instead of the roots A. The necessary and suflicient condition for stability is that each hypothetical single loop corresponding to a gain characteristic e# and feedback network A where x is a characteristic root of A, be stable in itself.

Now, if A is chosen so that A'=A, i. e., ai1'=au, for all z, a, then the relevant matrix is A A. As stated above, the characteristic roots of A A are all positive, if A is non-singular. Then the system can be universally stable, for all non-singular matrices A, if an is chosen equal to an, and if represents a stable feedback amplifier characteristic.

.To use the equation solver set the various controls at the values determined by the terms which they represent, it being noted that for each a network there is an a network having an identical setting. It follows therefore that the apparatus may be simplified by physically gauging the corresponding a and a networks. This greatly reduces the number of dials that need be set to solve a given equation. I

The actual value of the unknown terms may be determined in one of two ways:

(0.) Adjust the output of generator I to a value equal to a power of ten, positive or negative,

which is large enough to include the values of in turn, to the generator output to determine what fraction of the reference generator voltage plication of Brown and Goldberg referred to above. In this case no errors are introduced by errors in the voltmeter or changes in the amplitude of the reference voltage which may occur after it hasonce been set'tothe desired value.

This system greatly:simplifiesythe-zuse :of the apparatus, and is preferred, since it eliminates the necessity Ofi constantly;checking the input voltage and the voltmeter.

;In the earlier; system it, waspointed out that the ability to make a determinatiomis dependent '.uP 0I1=xthe-' system: itself-being stable, and this required the design of a special amplifier which had a phase tolerance ,not exceeding 90 throughout the range of frequencies for which the effective amplifier igain.

exceeded. unity but. .at, the same time the amplifier gain was required to be very large, a; factor normally contributing to instability. With a 1000 cycle gain'of about 15,000, in the earlier case, band pass filters were used. to attenuate the gain at, frequencies aboveiand below 1000 cycles, butto insure-the necessary phase tolerance it was nectessary-,,to :.consider the-operation of the amplifier :,between ;very Wide; frequency limits extending from a fractioniofrascycle,to nearly 10,000 c.

Moreover, stability Was ensured only for matrices whose characteristic -roots have positive real parts. In the present case the effective gain is the product of the ga-in'of two amplifiers. Con- -sequently" each amplifier may have a 7 much 1 smallernet gain at 1000 cycles than: was previous-- ly require'd. Thus-if ju= the gain of each amplifienneed now beequal to thesquare root of i that previously required. In the present case the effective gain of each-amplifier may be of the order of 100 to 125, say, a condition much easier The phase tolerance condition in the present:

cas.e.-is ;.:180 This" may. bedivided.:equal ;i90 .for each amplifier, cor one-may compensate for .excessive 'phase shift'of the: other.

However, since the gainisso low,no difficulty is encountered in meeting this, requirement. The :low gain,"

of course, greatly reduces the frequency range over which :it is necessary to consider thephase shift, although as before it. must be kept within :which the eiTective gain exceeds unity. :i-As a practhe limit stated above-for all frequencies for tical matter, conventional amplifiers can be used ,1, without difficulty since such amplifiers normally ,xmeet thGwphELSetQlEIflHOB limitation.

It, should'rbe understood, however, that ,u' need notbeequal-to In fact; either one of the ampli-- fiers may have-unitygain. As a practical matter .this-is not: preferred, however, because the high gain of the other amplifier-then -.increases the qdifficulty of phase control and no advantage, is

, t-he .za'; networks.

realizedwover the earlier system with regard to .vamplifierndesign.

A'- further i interesting feature 5 of the present invention is that the accuracy of solution'is subc -ing,;estab1is-hed.-. This-means that the ,only con- 1 tribution of the: a'inetworksjs toinsure stability,

, -whichnfact. substantiatesethe; theory above; since 1 theetransposeds-matri ;lnetworki-iwaseintnoduced onlyafor the ur-pose of insuring cimliversal stabilityi Asia practical matter ,theadvantagecof thisv factsis that, theme :potentiometers needr inot be accurate, but may be inexpfinsiveismall potentiometers ganged to:the respective accumtely calibrated: potentiometerswf :the.- .a. groups;;Aside from; the usecoili'additional,amplifiers; the cost iandesize Lof theeimproved equipment; isz;:increased but :slightly.

This. resultfimayabeiconfirmed by an; anakysispf Equations 131toi-18 base.d .on .theafact; that. when;

Q ,MM

is small, as it is, Equations 16 to 18 are substantiallyequal to zero. Use ismade oiia mathe- 'matical {theorem that when A isnon-singular the solution is independent of the values of a,

; whichr-is well .known and sneedcnot, be-explained herein.

Various modifications may be made in the.=apparatus described herein to illustrate the present invention. I In Fig.2; for example, the resistor 15 and the potentiometer l3 may conveniently be connected inithe plate and cathode circuits of an amplifier tube, thus eliminating the need of a coupling transformer. In fact such is the prejferred method-of ObtainingLtheHreference voltage which is described fully in the copending Brown and Goldberg application. In such case the-resister and potentiometer will not return to ground I directly but "toe suitable sources. ofz-iplatezvoltage and cathode bias, switching of these connections .being; accomplished by. additional contact arms on the polarity reversing-switch I I.

The apparatus described in the earlier Brown 1 and Goldberg. application i may i also be--usedi. in

practicing the present invention. This is explained in connection with -Flg. 3 which 'is a simplified block diagram of-the earlier device and to which reference is now made. Therleft hand column 'of' networks 9 are the four Y' networks of a device presumed to have been constructed in accordance with the earlier typecaof equation solver and having suihcient places toqsolve four simultaneous equations in four unknowns. Such a1; device-includes siXteen-additionat afl. networks arranged in four rows and four columns, and

, interconnected. aspshownt :The dashed lines indi catethatf or ,theipresent :use of ;the equipment the networks are to be grouped in fourgroups, each containing four networks. It will be understood that the interconnections of these networks and the four associated X amplifiers are "as described in the earlierapplicati'on. Notransposed a networks areincluded in-the original connection.

This arrangement 1 can be -used .in accordance with the present invention to solve two-simultaneous equations in two unknowns with universal stability provided .theggainiof each amplifier is reduced sufficiently to 'prevent oscillation, say by a factor of 10:1. To-solve ,the equationsqthe potentiometers of the upper two Y networks and the 3 potentiometers oflthe group 65 are all set at 0 as indicated by the "0 in each rectangle. The networks of group- 61 in the'upper righthand corner are set in accordance with the a coefficients of X, the firstrowurepresenting the coefficients of X1 and the second row representing the coefficients of X2. The twctlower networksare set to represent the Y terms of the two equations,

:astshown. The grouppfg-inetworks 1| in the lower left-hand corner are'set to represent the I es' pe d-te t e e ne t, tified as X1 J am lifi r 1 an "151 these amplifiers-are in the A ma rix feed into amp "fierX1 while'a z' and 0,22; amplifier'Xz. The ariip lifiers xr and'Xz feed the A matrix networks in 9Whih:fll;1@1nd:al2: eed back into amplifier X1, whilenetworks c121 and 1122 feed back into amplifier -X2. "Theother'netWOrks are not operative ,since no yoltage is developed :by theinetworkiwhen it set a-t fi ':.It"Was previ0Tus1yated= zhat say iof the order of ilo toolm the lmethodflofeusin the 2551 provide universal zstahi-lity teiie vely gconn ect theamhlifiers i SQ iQ 7h8d h nt aei be eofithe'order creeper unless 1h. gain is reduced. This should; he- .don s ab v WEYGI'M t .auve methedsissto us the networks:.,of igro'ups i back voltage to feither the the p'r pe haseito thereby hro, ucef99'% nerationi-or un twga he lonmxamn es ith diagonal netwerksE rand of reaped-lat, aximum output and-- egativegphasa: I erved th trthis -.-simply ppljfiSztC-rfifilfih fie the; full :ampl-ifler--vo1tae. ma nhas requ red t de enerateth igainrm tt ti 1 the X am- -in a ma n r-analo pu .to'th am je procedure o -ope ati -enablesun tmwnwith a coeff cient ;matrix-ral .z id' then *a ransposed co-' eflicient matrix -to :produqe tive of the unknown 'yalues. has beendesc i d wh ch:me 'he this i v tion to ether 'itv theseiieral mod f ations-thereof.

WhatIclaim is: p 1. "method of '-.determinhrg the u k wn terms -of 1a pluralityof of Jin-put voltagesirepres the c ta t ermsroi ea hee uatidm pe. amplifyingqsaid voltajggstby a factor l, deriving from each of said amplified voltagesaplurality of voltages approximatelytremesentative of the values of the coefficients of the respective unknowns; combining a voltage from each of said plurality of voltages with the voltage of every other plurality of voltages representative of the coefiicient of the same unknown; amplifying said combined voltages by a factor t to produce output voltages indicative of the values of the unknown terms of said equations, respectively; deriving from each of said output voltages a plurality of feedback voltages representative of the coefiicients of a given unknown; and combining with each input voltage the ones of said feedback voltages which represent the coefficients of the unknown terms of the equation associated with each constant term.

2. The method set forth in claim 1 which in- 75 I where Y 10 eludes the additional step of maintaining the overallampl'ification measured by the product oi and ,c' at a high value compared to the absolute values of the other terms of the equations to be solved. I

3. The method of solving n simultaneous equations of n unknown terms of the type when Y is a constant, X an unknown and aqthe coefficient of 'XQwhich includes the steps ,ofproviding n input voltages representative-"of" the values of "Y; amplifying said voltages by a factor p to; produce n first resultant voltages; deriving from each of said first resultant voltages n'coefflcient-representing voltages representativelof the coefficients of "X of a given equation; combining the coefiicient-representing voltages representing the coeflicients of the same unknown termsto produce n second resultant voltages; separately amplifying said second resultant voltages *by a factor to produce it output voltages indicative of the values of the unknown terms of said equations; deriving from each of said output voltages n feedback voltages representative of*t'he"co eificients of a given unknown term: and combining with said input voltages the ones ofsaid feedback voltages which represent the coefficients of the unknown terms of the equation assooiatedwith each-"constant term.

"4. The method of solving electronicallyaplurality of simultaneous equations which includes the steps of providing a first groupof voltages whose amplitudes'and phases are'to determine the Sign'and value of the unknownterms of the equations, respectively, deriving a second groupjof voltages'corresponding to the operation of a coeficientniatrix A on each'unknown term, combiningsaid second-group of voltages with-additional voltages representing, respectively, the-constant terinsof said equations, amplifyingsaid combined voltagesjto produce athird groupof voltages; operating on said third group of voltages'v'vitha coefficient matrix A to'produce a fourth group f-voltages, and amplifying said fourth-group-of voltagesthe latter-amplified voltages constituting the first group of voltages, and rnea su'rin'g'said last-named voltages to determine-the values-of the unknown terms of said equations.

5. The method of determining the vaiuesm the unknown-terms of n si multaneous equations of the type. I

' F u' r-d' n g-I -Y 21X1+-G22 X2+ 27px" is a constant, a is a coefficient of X and X an unknown, which includes the steps of providing n input voltages representative of the values of Y1, Y2 to Yn; separately amplifying said input voltages, deriving from each amplified voltage n coefficient-representing voltages whose polarities and amplitudes are determined by the coefiicients of X of the respective equations; combining the coeiiicient-representing voltages which represent the various coefficients of the same unknown to produce 12 resultant voltages; separately amplifying said resultant voltages, the amplitudes and phases of said amplified resultant voltages being proportional to the values of said unknown terms X1, X2 and Xn respectively; derivl 1 ing from each of said amplified resultant voltages 12- feeolback voltages representative of the n coefficients of X1, X2 and Xn respectively; and combining with said input voltages the ones of said feedback voltages which represent the coefficients of the unknown terms of the equation associated with each constant term.

6. A device of the character described which includes a source of voltage, a first plurality of voltage dividers connected to said source; a first group of amplifiers connected one to each voltage divider, a second plurality of voltage dividers connected to the output of each amplifier, the output of each voltage divider of a given amplifier being coupled to one voltage divider of each other amplifier and to one of a second group of amplifiers; a third plurality of voltage dividers connected to the output of each amplifier of the second group, the output of each voltage divider of a given amplifier of the second group being coupled to the output of one voltage divider of each other amplifier of the second group and to the input of one of the first group of amplifiers.

7. A device of the character described in claim 6 in which the voltage dividers of both groups are coupled together through resistors.

8.. A device of the character described in claim 6 in which the product of the gain of the amplifiers of the first group multiplied by the gain of the amplifiers of the second group is of the order of ten thousand.

9. A device of the character described in claim 6 in which there is associated with each voltage divider means for reversing the phase of the output voltage produced thereby.

10. An electronic simultaneous equationsolver comprising a source of voltage; a plurality of networks connected to said source, a like number of input amplifiers, the output of each network being resistance coupled to the input of one 01 said amplifiers; a second group of networks, a plurality of said networks being connected to the output of each input amplifier; a like number of output amplifiers, the input of each output amplifier being coupled by a resistance to one of the networks of each input amplifier; and a third group of networks connected to the output of each output amplifier, one network of each output amplifier being coupled to each of said input amplifiers.

11. An electronic simultaneous equation solver which comprises a first group of voltage'dividers connected to a source of reference potential and adapted to be set to represent the constant terms of a system of non-singular simultaneous equations, a second group of voltage dividers adapted to be set to represent the coeflicients of the unknown terms of the system of equations, a third group of voltage dividers individually coupled to respective ones of said second group of voltage dividers; and means including amplifiers interconnecting said networks whereby output voltages are obtained which are proportional to the unknown terms of said system of equations independently of the nature of its characteristic roots.

12. A device of the character described in claim 10 in which each network of said third group is mechanicallycoupled to and adjustable with one network of saidsecond group.

13. A device for electronically determining the values of the unknown terms of a system of n simultaneous equations of the type where Y1, Y2 Yn are constant terms, X1, X2. .Xn the unknown terms and a the coeflicients of the unknown terms which are identified by the second'subscript in the equations which are identified by the first subscript, which includes means for producing input voltages representative of the constant terms Y1, Y2. .Yn; an input amplifier for separately amplifying each of said voltages; a first plurality of voltage dividing networks connected to each input amplifier, said networks being adapted to be set at least approximately to the fractional values of the amplified voltages Y1, Y2. .Yn which are determined by the values of the a. coefficients whose first subscripts are I, 2, .11., respectively; means for combining the voltages representing the a coeflicients'whose second-subscripts are I, 2,. .12, respectively to produce 11 resultant voltages; an output amplifier for separately amplifying each of said resultant voltages;'a second plurality of voltage dividingnetworks connected to each output amplifier, the latter network being adapted to be set to the fractional values of the amplified voltages Y1, Yz .Yn which are determined by the values'of the a. coefficients whose second subscripts are I, 2, n, respectively; means for'combining the voltages developed representing the a coefficients whose'first subscripts are I, 2, .12, respectively, to produce n feedback voltages; and means for combining said feedback voltages with respective ones of said input voltages; the amplitudes of the amplified resultant voltages being indicative of the value of said unknown terms. t

14. A device of the character described in claim 13 in which the networks of said first plurality are gangedwith' the networks of the second plurality which are to be set to represent coeflicientvoltages' identified by the same subscripts.

P GEORGE W. BROWN.

No references cited. 

